for a zero mean error vector \(\)  with independent entries. The entries of the \(d \times K\)  matrix \(\)  are given by \(A_{jk}=\lbrace j\in G^*_{k}\rbrace \) . A cluster of variables consist in those components of \(\)  with indices in the same \(G^*_{k}\) . We denote \(()=^*\) , a diagonal matrix with entries \(\Gamma ^*_{jj} =\gamma _{j}^*\)  for any \(1\le j\le d\) . We also assume that the mean-zero noise \(\)  is independent of \(\) . The clusters are uniquely defined by model (REF ) provided that the smallest cluster contains at least two variables and that \(_{  }^*\)  is strictly positive definite,
as shown in [1]}, and this result holds irrespective of distributional assumptions.
To keep the presentation focused, in this work we assume that \(\sim (0,_{  }^*)\)  and \(\sim (0,^*)\) , which implies \(\sim (0,_{  }^*)\)  with \(_{  }^*=_{  }^*^T+^*\) . In this context we consider two related, but different, graphical models:
